1. Field of the Invention
The present invention relates to numerical modeling and simulation methods, and particularly to a computerized method of modeling residual stresses during laser cutting that utilizes thermal diffusion and stress equations and a discretization numerical method to model temperature variation and residual stresses in a substrate material due to laser cutting therethrough of small diameter holes.
2. Description of the Related Art
In continuum mechanics, the concept of stress, introduced by Cauchy around 1822, is a measure of the average amount of force exerted per unit area of the surface on which internal forces act within a deformable body. In other words, stress is a measure of the intensity, or internal distribution, of the total internal forces acting within a deformable body across imaginary surfaces. These internal forces are produced between the particles in the body as a reaction to external forces applied on the body. External forces are either surface forces or body forces. Because the loaded deformable body is assumed as a continuum, these internal forces are distributed continuously within the volume of the material body; i.e. the stress distribution in the body is expressed as a piecewise continuous function of space coordinates and time.
For the simple case of a body which is axially loaded (e.g., a prismatic bar subjected to tension or compression by a force passing through its centroid), the stress σ, or intensity of the distribution of internal forces, can be obtained by dividing the total tensile or compressive force F by the cross-sectional area A upon which it is acting, or
  σ  =            F      A        .  
In this simplified case, the stress σ is represented by a scalar called engineering stress or nominal stress that represents an average stress over the area; i.e., the stress in the cross section is uniformly distributed. In general, however, the stress is not uniformly distributed over a cross section of a material body, and consequently the stress at a point on a given area is different than the average stress over the entire area. Therefore, it is necessary to define the stress not at a given area but at a specific point in the body. According to Cauchy, the stress at any point in an object, assumed to be a continuum, is completely defined by the nine components σij of a second order tensor known as the Cauchy stress tensor.
The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point, the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible. In the three-dimensional space of the principal stresses (σ1, σ2, σ3), an infinite number of yield points form together a yield surface.
Knowledge of the yield point is vital when designing a component, since it generally represents an upper limit to the load that can be applied. It is also important for the control of many materials production techniques such as forging, rolling, or pressing. In structural engineering, this is a soft failure mode, which does not normally cause catastrophic failure or ultimate failure unless it accelerates buckling.
The laser cutting process finds wide applications in industry, due to its precise operation, rapid processing, and the low heat-affected zone generated around the cut edges. The temperature rise in the irradiated region generates molten metal, and removal of the molten metal from the workpiece forms the cut edges. Although the temperature rise in the molten metal is higher than the melting temperature of the substrate material, temperature remains almost at the melting temperature at the kerf edges. This is due to heat transfer between the molten metal and the substrate material at the solid-liquid interface. Consequently, the cut edges cannot extend further into the solid substrate, and the size of the kerf width remains almost constant during the cutting process. Although temperature at the cut edge in the region of the laser-irradiated spot remains at the melting temperature of the substrate material, high temperature gradients within the neighboring solid phase can result. The high temperature gradient around the cut edges results in the development of thermal stresses during the cutting process.
Since these thermal stress levels exceed the elastic limit of the substrate material, residual stresses are developed along the cutting edge. This situation becomes severe for the cutting of relatively small-diameter holes in the substrate, which, in turn, influences the quality of the end product. Consequently, modeling and simulation of the laser cutting of small-diameter holes and the residual stress developed around the cut edges becomes essential.
Thus, a method of modeling residual stresses during laser cutting solving the aforementioned problems is desired.